Real-space finite-difference calculation method of generalized Bloch wave functions and complex band structures with reduced computational cost

Abstract

Generalized Bloch wave functions of bulk structures, which are composed of not only propagating waves but also decaying and growing evanescent waves, are known to be essential for defining the open boundary conditions in the calculations of the electronic surface states and scattering wave functions of surface and junction structures. Electronic complex band structures being derived from the generalized Bloch wave functions are also essential for studying bound states of the surface and junction structures, which do not appear in conventional band structures. We present a novel calculation method to obtain the generalized Bloch wave functions of periodic bulk structures by solving a generalized eigenvalue problem, whose dimension is drastically reduced in comparison with the conventional generalized eigenvalue problem derived by Fujimoto and Hirose [Phys. Rev. B 67, 195315 (2003)]. The generalized eigenvalue problem derived in this work is even mathematically equivalent to the conventional one, and, thus, we reduce computational cost for solving the eigenvalue problem considerably without any approximation and losing the strictness of the formulations. To exhibit the performance of the present method, we demonstrate practical calculations of electronic complex band structures and electron transport properties of Al and Cu nanoscale systems. Moreover, employing atom-structured electrodes and jellium-approximated ones for both of the Al and Si monatomic chains, we investigate how much the electron transport properties are unphysically affected by the jellium parts.

Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations

“…However, an additional computational cost is incurred when we use the BiCG method for solving Eq. (25). If we use the BiCG method as the solver, it is difficult to set a reasonable number of right-hand side N rh owing to a variation in the eigenvalue distribution inside the ring-shaped region.…”

“…To demonstrate speed-ups, we compare the computational time of our method with that of the OBM method, which is categorized as a WFM method. Although continuous improvements [24][25][26]56 have been made after the first study of the OBM method, the computation of the first and last r columns of (E − H l,l ) −1 and the 2r-dimensional generalized eigenvalue problem is still required. In this study, the matrix inversion is calculated using the CG method, 57 and the generalized eigenvalue problem is solved by the optimized LAPACK routine ZGGEV and SS method.…”

Self-energy matrices for electron transport calculations within the real-space finite-difference formalism

“…Hence, they are given as integer multiples of the number of the grid points in an xy plane (for more details, see Ref. [35]). The subscript L (R) denotes that a quantity with the subscript is associated with the left (right) boundary.…”

“…(8) in Ref. [35], one can see that the sets of generalized Bloch wave functions in the boundary regions, Q i,M(N) = {q i M(N),1 ,q i M(N),2 , . .…”

Improved recursive Green's function formalism for quasi one-dimensional systems with realistic defects

“…Different solutions have been found, e.g. a pivoting-like bandwidth minimization method in combination with a block-tridiagonalization algorithm [Wim09], the reverse Cuthill-McKee algorithm for the block-tridiagonalization of connected graphs [Mas11,Cut69], the Knitting algorithm, which uses the Dyson equation and the resulting decimation scheme to build up the system site by site [Kaz08], or accelerations by using singular value decompositions [Tsu14]. Besides the RGF, also other methods are often utilized.…”